Let $F_n$ denote the collection of all the $2^n$ many piecewise linear continuous functions $f:[0,1]\rightarrow\mathbb R$ such that $f(0)=0$ and $f$ is linear with slope $\pm \sqrt{n}$ on the intervals $[\frac in,\frac{i+1}n]$ for $0\le i<n$.

Let $\phi_n$ denote a uniformly randomly chosen element of
$$
\text{arg min}_{f\in F_n}\left(\sup_{0\le x\le 1}|W(x)-f(x)|\right).
$$
In other words, $\phi_n$ is an element of $F_n$ that minimizes the sup-norm distance to $W$. More simply, we can say that $\phi_n$ is a nearest walk to Brownian motion.

Are you just trying to create some sequence of random walks that converges to $W$ uniformly almost surely or you are more ambitious than that?
–
fedjaSep 12 '10 at 16:20

@fedja: in that case I guess there is the Skorokhod embedding which you may be thinking of. The sequence of walks should be close to uniform and converge at a close to optimal rate for the application I have in mind (which is to answer Question 5.1 in my paper math.hawaii.edu/~bjoern/Publications/… with Szabados).
–
Bjørn Kjos-HanssenSep 12 '10 at 16:50